Found problems: 85335
2015 Vietnam National Olympiad, 1
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and
${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$
Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.
1996 Bosnia and Herzegovina Team Selection Test, 2
$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function
$b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$
1988 Bulgaria National Olympiad, Problem 4
Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point.
2023 Malaysian IMO Training Camp, 3
A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$
[i]Proposed by Ivan Chan Kai Chin[/i]
2008 Indonesia MO, 3
Find all natural number which can be expressed in
$ \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b}$
where $ a,b,c\in \mathbb{N}$ satisfy
$ \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1$
2023 Kyiv City MO Round 1, Problem 4
Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds:
$$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$
For which largest $k$ do there exist $k$-similar numbers?
[i]Proposed by Oleksiy Masalitin[/i]
2021 China Second Round A1, 1
In triangle ABC,X,Y are on the angle bisector of ∠BAC and ∠ABX=∠ACY.BX intersects CY at P and circles (BYP) and (CXP) intersect at Q different from P. Prove that A,P,Q are on a line.
2002 AMC 12/AHSME, 24
A convex quadrilateral $ ABCD$ with area $ 2002$ contains a point $ P$ in its interior such that $ PA \equal{} 24$, $ PB \equal{} 32$, $ PC \equal{} 28$, and $ PD \equal{} 45$. FInd the perimeter of $ ABCD$.
$ \textbf{(A)}\ 4\sqrt {2002}\qquad \textbf{(B)}\ 2\sqrt {8465}\qquad \textbf{(C)}\ 2\left(48 \plus{} \sqrt {2002}\right)$
$ \textbf{(D)}\ 2\sqrt {8633}\qquad \textbf{(E)}\ 4\left(36 \plus{} \sqrt {113}\right)$
1998 Austrian-Polish Competition, 1
Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$
2009 Italy TST, 3
Find all pairs of integers $(x,y)$ such that
\[ y^3=8x^6+2x^3y-y^2.\]
2002 Korea - Final Round, 2
Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.
2016 Dutch BxMO TST, 1
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.
1976 Dutch Mathematical Olympiad, 3
In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?
2012 Dutch BxMO/EGMO TST, 5
Let $A$ be a set of positive integers having the following property:
for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$.
Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.
2017 ASDAN Math Tournament, 5
Compute
$$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$
2010 Postal Coaching, 3
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$\boxed{1} \ f(1) = 1$
$\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$
1979 Romania Team Selection Tests, 5.
In how many ways can we fill the cells of a $m\times n$ board with $+1$ and $-1$ such that the product of numbers on each line and on each column are all equal to $-1$?
1990 IMO Longlists, 14
We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.
2005 CHKMO, 2
In a school there $b$ teachers and $c$ students. Suppose that
a) each teacher teaches exactly $k$ students, and
b)for any two (distinct) students , exactly $h$ teachers teach both of them.
Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.
2007 iTest Tournament of Champions, 5
Let $c$ be the number of ways to choose three vertices of an $6$-dimensional cube that form an equilateral triangle. Find the remainder when $c$ is divided by $2007$.
2020 CMIMC Combinatorics & Computer Science, 4
The continent of Trianglandia is an equilateral triangle of side length $9$, divided into $81$ triangular countries of side length $1$. Each country has the resources to choose at most $1$ of its $3$ sides and build a “wall” covering that entire side. However, since all the countries are at war, no two countries are willing to have their walls touch, even at a corner. What is the maximum number of walls that can be built in Trianglandia?
2025 Sharygin Geometry Olympiad, 19
Let $I$ be the incenter of a triangle $ABC$; $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the orthocenters of the triangles $BIC$, $AIC$, $AIB$; $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of $BC$, $CA$, $AB$, and $S_{a}$, $S_{b}$, $S_{c}$ be the midpoints of $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$. Prove that $M_{a}S_{a}$, $M_{b}S_{b}$, $M_{c}S_{c}$ concur.
Proposed by: S Kuznetsov
2002 Germany Team Selection Test, 3
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
1998 South africa National Olympiad, 3
$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.
PEN H Problems, 4
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.